Some explicit arithmetic on curves of genus three and their applications

TL;DR

This paper develops explicit arithmetic algorithms for genus 3 curves, including formulas for Howe curves and methods to compute decomposed Richelot isogenies, aiding classification and analysis of these algebraic structures.

Contribution

It introduces explicit formulas and algorithms for genus 3 curves, including Howe curves and Richelot isogenies, advancing computational methods in algebraic geometry.

Findings

Algorithms to compute the codomain of decomposed Richelot isogenies.

Explicit formulas for defining equations of Howe curves.

Complexity analysis for enumerating superspecial Howe curves.

Abstract

A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this paper, we develop some explicit arithmetic on curves of genus $3$, including algorithms to compute the codomain of a decomposed Richelot isogeny. As solutions to compute the domain of a decomposed Richelot isogeny, explicit formulae of defining equations for Howe curves of genus $3$ are also given. Using the formulae, we shall construct an algorithm with complexity $\tilde{O}(p^3)$ (resp. $\tilde{O}(p^4)$) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus $3$.